ARTICLE IN PRESS. Giuseppe Mitri a,b,,adamp.showman a

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1 S (05) /FLA AID:7639 Vol. ( ) [DTD5] P.1 (1-14) YICAR:m5 v 1.39 Prn:25/05/2005; 7:41 yicar7639 by:diana p. 1 Icarus ( ) Convective conductive transitions and sensitivity of a convecting ice shell to perturbations in heat flux and tidal-heating rate: Implications for Europa Giuseppe Mitri a,b,,adamp.showman a a Department of Planetary Sciences, Lunar and Planetary Laboratory, University of Arizona, Tucson, AZ, USA b International Research School of Planetary Sciences, University G. d Annunzio, Pescara, Italy Received 17 July 2004; revised 29 January 2005 Abstract We investigate the response of conductive and convective ice shells on Europa to variations of heat flux and interior tidal-heating rate. We present numerical simulations of convection in Europa s ice shell with Newtonian, temperature-dependent viscosity and tidal heating. Modest variations in the heat flux supplied to the base of a convective ice shell, F, can cause large variations of the ice-shell thickness δ. In contrast, for a conductive ice shell, large F involves relatively small δ. We demonstrate that, for a fluid with temperature-dependent viscosity, the heat flux undergoes a finite-amplitude jump at the critical Rayleigh number Ra cr. This jump implies that, for a range of heat fluxes relevant to Europa, two equilibrium states corresponding to a thin, conductive shell and a thick, convective shell exist for a given heat flux. We show that, as a result, modest variations in heat flux near the critical Rayleigh number can force the ice shell to switch between the thin, conductive and thick, convective configurations over a year interval, with thickness changes of up to km. Depending on the orbital and thermal history, such switches might occur repeatedly. However, existing evolution models based on parameterizedconvection schemes have to date not allowed these transitions to occur. Rapid thickening of the ice shell would cause radial expansion of Europa, which could produce extensional tectonic features such as fractures or bands. Furthermore, based on interpretations for how features such as chaos and ridges are formed, several authors have suggested that Europa s ice shell has recently undergone changes in thickness. Our model provides a mechanism for such changes to occur Elsevier Inc. All rights reserved. Keywords: Europa; Geophysics; Ices; Interiors; Satellites of Jupiter; Thermal convection; Thermal histories 1. Introduction We investigate the response of a conductive and convective ice shell to changes of heat production in the silicate mantle and in the ice shell of Jupiter s satellite Europa. Estimates of Europa s ice-shell thickness range from 3 to 50 km (Turtle and Ivanov, 2002; Schenk, 2002; Hussmann et al., 2002; Tobie et al., 2003; Williams and Greeley, 1998; Greenberg et al., 1998, 1999). This uncertainty in thickness * Corresponding author. addresses: mitri@lpl.arizona.edu (G. Mitri), showman@lpl.arizona.edu (A.P. Showman). translates directly into an uncertainty in the heat-transfer mechanism: if the shell is thick, the rigid surface could be underlain by a layer of convecting water ice (Cassen et al., 1979; McKinnon, 1999; Pappalardo et al., 1998; Showman and Han, 2004; Tobie et al., 2003), whereas a thin shell would instead transport the heat by conduction (Greenberg et al., 1998, 1999). Interestingly, most estimates of the shell thickness (10 40 km) imply that the ice-shell Rayleigh number is near the critical Rayleigh number, which is 10 6 for realistic temperature-dependent viscosities (e.g., McKinnon, 1999). The implications of surface landforms for the configuration of the ice shell remain controversial (Pappalardo et /$ see front matter 2005 Elsevier Inc. All rights reserved. doi: /j.icarus

2 S (05) /FLA AID:7639 Vol. ( ) [DTD5] P.2 (1-14) YICAR:m5 v 1.39 Prn:25/05/2005; 7:41 yicar7639 by:diana p. 2 2 G. Mitri, A.P. Showman / Icarus ( ) Fig. 1. Structures and temperature profiles for two possible configurations of the ice shell of Europa. The heat generated in the interior of the planet might be shed by simple conduction through a relatively thin ice shell directly overlying a subsurface ocean (A), or transported equally well through a much thicker ice shell containing of an actively convecting layer (B). Because of the large viscosity near the surface, the ice shell in B develops a stagnant lid at the surface; the convective motions are confined to a sublayer of the ice shell. The states (A) and (B) could have very similar heat fluxes. al., 1999). Numerous small (3 30-km diameter) pits, uplifts, and disrupted spots, as well as larger chaos terrains such as Conamara Chaos and the Mitten, have been attributed to convection in an ice shell at least 10-km thick (Pappalardo et al., 1998; Head and Pappalardo, 1999; Collins et al., 1999, 2000; Figueredo et al., 2002). But other authors have suggested that chaos results instead from melt-through of a thin ice shell (Williams and Greeley, 1998; Greenberg et al., 1999, 2003; O Brien et al., 2002; Melosh et al., 2004). Similarly, some formation mechanisms for ridges require a thin-ice shell (Greenberg et al., 1998), while other ridgeformation mechanisms allow a thicker shell (Melosh and Turtle, 2004; Nimmo and Gaidos, 2002). Figueredo and Greeley (2004), Prockter et al. (1999), Pappalardo et al. (1999), and others have shown that tectonic resurfacing (i.e., ridge building) decreased rapidly after ridged-plains formation and that chaos formation has increased with time. These authors suggest that the transition from tectonic- to chaos resurfacing resulted from the gradual thickening of the ice shell. On the other hand, Greenberg et al. (1999, 2000, 2003) suggests a different scenario where the chaos and tectonic terrains form concurrently and continually resurface Europa. Fig. 1 shows the schematic structures and temperature profiles for the two plausible configurations of the ice shell (conductive and convective). Both structures have a steep temperature gradient within the top layer and lose heat conductively through this layer. Within the conductive regime, heat flux follows a rough inverse proportionality to layer thickness, so thicker layers transport smaller heat fluxes. In layers thicker than a critical value, δ crit, the Rayleigh number exceeds the critical Rayleigh number, and convection begins. The assumption is often made that the heat flux near this transition is a continuous function of the layer thickness. In this case, the heat flux of the conductive solution (A) would exceed that of the convective solution (B). However, laboratory experiments in a fluid with temperaturedependent viscosity indicate that, at the critical Rayleigh number, the convection jumps directly to a finite-amplitude regime (Stengel et al., 1982), implying that the heat flux for a convective layer infinitesimally thicker than δ crit exceeds that for a conductive layer infinitesimally thinner than δ crit. This implies that the heat flux for a convective layer infinitesimally thicker than δ crit will be equal to the heat flux for a conductive state that is substantially thinner than δ crit. Therefore, for a range of conditions near the critical Rayleigh number, two solutions one a thin, conductive shell and the other a thick, convective shell exist for a given basal heat flux. The existence of two solutions for a given heat flux raises an obvious question: What determines which of the two states Europa occupies? And can Europa switch between these states? Furthermore, in the convective regime, how sensitive is the ice-shell thickness to perturbations in the basal heat flux and internal tidal-heating rate? Answers to these questions have important implications for the time history of Europa s ice-shell thickness, and hence for Europa s surface geology. These questions become even more relevant considering that Europa s heat-production rate may vary in time. Io s measured heat flux exceeds that possible in steady state, which may result from 10 8 year oscillations in Io s eccentricity and tidal-heating rate caused by coupled orbitalgeophysical feedbacks (Greenberg, 1982; Ojakangas and Stevenson, 1986). Similar feedbacks may be relevant for Ganymede (Showman et al., 1997) and Europa (Hussmann and Spohn, 2004). (In any case, oscillations of Io s eccentricity would cause oscillations in Europa s eccentricity and tidal-heating rate even in the absence of any feedbacks in Europa.) The observational suggestion that Europa s geology has shifted from ridge to chaos formation over the past 50 Myr (Pappalardo et al., 1999; Prockter et al., 1999; Greeley et al., 2000; Kadel et al., 2000; Figueredo and Greeley, 2004) supports the idea that variations in Europa s internal heating rate and ice-shell thickness have occurred over time. Here we present two-dimensional numerical simulations of convection in Europa s ice shell, including tidal heating, to investigate the sensitivity of the ice-shell thickness to perturbations in basal heat flux and interior tidal-heating rate. Our simulations confirm Stengel et al. s (1982) result that the heat flux undergoes a finite-amplitude jump at the critical Rayleigh number; furthermore, this occurs at heat fluxes relevant to Europa ( W m 2 depending on the viscosity and tidal-heating rate). We determine the amplitude of the heat-flux jump and discuss the consequences for the thermal evolution of the ice shell. We show that, if the system is near the critical Rayleigh number, small changes in the heat flux can force the system to switch from conductive to convective configurations with large variations in

3 S (05) /FLA AID:7639 Vol. ( ) [DTD5] P.3 (1-14) YICAR:m5 v 1.39 Prn:25/05/2005; 7:41 yicar7639 by:diana p. 3 Europa convection 3 thickness of the ice shell. The consequent volume changes of Europa can generate surface stresses up to 100 bar, potentially causing formation of fractures, bands, or other structures. The direct jump to finite-amplitude convection also increases the likelihood that convection can disrupt the surface even when the Rayleigh number is barely supercritical, which may help to allow the production of chaos, pits, and uplifts soon after initiation of convection in a gradually thickening ice shell (as has been suggested by several authors). The models presented here may also have applicability for the thermal evolution of Ganymede, Callisto, Titan, and the small icy moons of Saturn and Uranus. 2. Model We consider a model of an ice shell overlying a global ocean. The Boussinesq fluid equations are solved with the ConMan finite-element code (King et al., 1990). The code solves the dimensionless equations of momentum, continuity and energy, respectively given by (e.g., Schubert et al., 2001) σ ij + Raϑk i = 0, x j u i = 0, x i ϑ t ϑ + u i = 2 ϑ x i xi 2 + Q, where σ ij is the dimensionless stress tensor, u i is the dimensionless velocity, ϑ is dimensionless temperature, k i is the vertical unit vector, t is dimensionless time, x i and x j are the dimensionless spatial coordinates, and i and j are the coordinate indices. Q is the dimensionless volumetric heating rate. The nondimensionalization is performed using length, speed, and time scales of δ, κ/δ, and δ 2 /κ, respectively, where δ is the ice-shell thickness, and κ is the thermal diffusivity. The stresses in dimensional form are given by ( ui σ ij = pδ ij + η x j + u j x i ), where p is the pressure, δ ij is the Kronecker delta, and η is the temperature-dependent viscosity. The Rayleigh number Ra is given by Ra = α iceρg T δ 3, (5) κη 0 where g is gravitational acceleration (1.32 m s 2 ), α ice is the coefficient of thermal expansion ( K 1 ), κ is the thermal diffusivity of the water ice ( m 2 s 1 ), ρ is the density of the water ice (917 kg m 3 ), T is the temperature difference between the bottom and the top surfaces, and η 0 is the viscosity of the water ice at the melting point (1) (2) (3) (4) Table 1 Physical parameters Thermal expansivity of ice α ice K 1 Gravitational acceleration g 1.32 m s 2 Thermal diffusivity of ice κ m 2 s 1 Thermal conductivity of ice k 2.3 W m 1 K 1 Ice density ρ 917 kg m 3 Surface temperature T s 95 K Melting temperature of ice T m 270 K Ice viscosity at T m η Pa s Europa radius R 1569 km Orbital frequency ω s 1 Rigidity of ice µ Pa ( Pa s). The physical parameters used in the model are summarized in Table 1. The surface temperature is held at T s = 95 K and the bottom temperature of the ice shell is fixed at the melting temperature of fresh water, T m = 270 K. The numerical simulations are performed in a Cartesian domain, which is a good approximation for Europa s ice shell because its thickness is much less than the radius of the satellite. Most simulations used a resolution of finite elements, but the resolution was increased to elements for the high Rayleigh number cases. This improved resolution is needed to resolve the fine-scale convective structure, and obtain proper estimates of the heat flux, for the high-rayleigh-number cases. The highest-rayleigh-number cases (Ra = 10 9 ) were rerun at to verify numerical convergence of the heat fluxes. The velocity boundary conditions are periodic on the sides and free-slip rigid walls on the top and the bottom of the domain. The domain has a width:height ratio of 4:1, which is large enough to minimize the influence of the domain on the convective structure. Selected cases were also rerun at a width:height ratio of 6:1 to verify that (consistent with the findings of Solomatov and Moresi, 2000) the domain has minimal influence on the results. The simulations are initialized using a conductive temperature profile containing a sinusoidal thermal perturbation with an amplitude of K. Goldsby and Kohlstedt (2001) proposed a composite constitutive equation to account for the combined strain rate ε of several deformation mechanisms as follows: ( 1 ε = ε diff ) 1 + ε disl, (6) ε basal ε gbs where ε diff, ε basal, ε gbs, ε disl are strain rate of diffusional flow, basal slip, grain boundary sliding, and dislocation creep, respectively. The strain rate for a single mechanism ε j can be described by ε j = A σ n d p exp ( Q RT ), where d is the grain size, σ is the shear stress, A, n, and p are constants, Q is the activation energy, R is the gas constant, and T is temperature (Goldsby and Kohlstedt, 2001). The flow law parameters are in Table 2. The diffusion creep (7)

4 S (05) /FLA AID:7639 Vol. ( ) [DTD5] P.4 (1-14) YICAR:m5 v 1.39 Prn:25/05/2005; 7:41 yicar7639 by:diana p. 4 4 G. Mitri, A.P. Showman / Icarus ( ) Table 2 Flow law parameters (Goldsby and Kohlstedt, 2001) Creep regime A n Q (kj mol 1 ) p Dislocation creep (T < 258 K) MPa 4.0 s Dislocation creep (T > 258 K) MPa 4.0 s GBS (T < 255 K) MPa 1.8 m 1.4 s GBS (T > 255 K) MPa 1.8 m 1.4 s Basal slip MPa 1.8 s rate is calculated using the equation (Goldsby and Kohlstedt, 2001) ε = 42σV ( m (8) RTd 2 D v + πδ ) g d D b, where V m is the molar volume ( m 3 ), δ g is the grain boundary width ( m), and D v and D b are volume and grain boundary diffusion coefficients, respectively, each of the form D i = D 0,i exp( Q RT ). Goldsby and Kohlstedt (2001) assumed that Q = 49 kj mol 1 and the preexponent term for grain boundary diffusion is equivalent to that for volume diffusion ( m 2 s 1 ). The predominant creep mechanisms under conditions relevant to a convecting ice shell are superplastic deformation with grain boundary sliding (GBS) in combination with diffusion motion (Table 3) (McKinnon, 1999; Goldsby and Kohlstedt, 2001; Durham et al., 1997). McKinnon (1998, 1999) argued that the thermal-buoyancy stress is 0.1ρgα T plume δ, where T plume 5 K is the temperature difference between warm and cold plumes. For Europa, this scaling implies convective stresses of MPa, whereas the expected tidal stress MPa, depending on the ice-shell temperature. McKinnon (1999) suggested that if σ conv σ tid, the tidal stresses act to linearize the viscosity, eliminating the dependence of viscosity on the convective stress even for non-newtonian creep mechanisms. Nevertheless, the stresses are uncertain and it is possible that convective and tidal stresses are comparable. Even in this case, however, the fact that GBS is only mildly non-newtonian implies that the viscosity varies by only a factor of 2 5 with plausible variations in stress, which is much smaller than the ordersof-magnitude viscosity variations associated with changes in temperature (e.g., Showman and Han, 2004). Consequently, here we assume that the viscosity is Newtonian, and we defer exploration of non-newtonian rheologies to future work. The Newtonian temperature dependent viscosity of the water ice is given by [ ( )] η = η 0 exp A Tm (9) T 1. We use a value of the constant A = 7.5, which yields a viscosity contrast η, defined as the ratio between the maximum and the minimum viscosities in the simulation, of η = This value is chosen to maintain the convection in the stagnant-lid regime (e.g., Solomatov and Moresi, 2000) while ensuring that the viscosity contrast across each finite element is less than a factor of 2, as necessary to maintain numerical accuracy (Moresi et al., 1996). Although the actual viscosity contrast far exceeds 10 6, we expect that our simulations preserve the essential aspects of the convective behavior. Larger values of A would lead to larger values of the critical Rayleigh number, which would increase the minimum ice-shell thickness for convection to initiate. The qualitative existence of a heat-flux jump at the critical Rayleigh number should not be affected, however (in fact, we expect that our simulations at artificially small η underestimate the amplitude of the heat-flux jump at the critical Rayleigh number). For this preliminary study, we ignore the pressuredependence of the melting temperature, which is approximately valid for ice shells thinner than 50 km, relevant to Europa. Pressure-dependence of the melting temperature is more likely to play a role for satellites with thicker ice shells such as Ganymede, Callisto, or Titan. The possible presence of salts and ammonia in the ocean may reduce the melting temperature of the ice by up to a few K for plausible compositions (e.g., Kargel, 1991); however, in this model we neglect this effect and maintain the basal temperature at 270 K. The thermal conductivity of ice depends on the temperature, but ConMan assumes constant conductivity. The convecting sublayer remains nearly isothermal, and most of the temperature variation occurs within the stagnant lid. We therefore expect that constant-conductivity simulations provide reasonably accurate solutions throughout the convection region; the primary effect of variable conductivity Table 3 Strain rates of diffusion flow, basal slip, grain boundary sliding, dislocation creep, and the total strain rate using the composite constitutive equation for different grain size of the water ice in the tidal stress range ( Pa). The creep mechanism that likely contributes most to the strain rate on Europa conditions of the ice shell is the superplastic flow regime with grain boundary sliding (GBS) deformation process in combination with diffusion motion Grain size m m m m Stress 10 4 Pa 10 5 Pa 10 4 Pa 10 5 Pa Dislocation creep GBS Basal slip Diffusion Total

5 S (05) /FLA AID:7639 Vol. ( ) [DTD5] P.5 (1-14) YICAR:m5 v 1.39 Prn:25/05/2005; 7:41 yicar7639 by:diana p. 5 Europa convection 5 would be to alter the thickness of the stagnant lid by 50% for a given heat flux (see Tobie et al., 2003 for discussion). We include temperature-dependent internal tidal heating in the numerical model. We use the volumetric dissipation rate for an incompressible Maxwell body, which can be written q = ε2 0 ω2 η, [1 + ω2 η 2 ] µ 2 (10) where ω = s 1 is the Europan orbital frequency, µ = Pa is the rigidity of the water ice, and ε 0 is the maximum magnitude of tidal-flexing strain of the ice shell during Europa s orbit around Jupiter. The temperature dependence of q is determined by the temperature dependence of the ice viscosity given by Eq. (9). A simple estimate of ε 0 is given by the ratio ξ/r between the amplitude of the vertical surface deformation ξ during a tidal cycle (20 30 m) (Ross and Schubert, 1987; Moore and Schubert, 2000) and the radius of Europa (R = 1569 km). The tidal flexing amplitude ε 0 ranges from to (Showman and Han, 2004). At high Rayleigh numbers, the heat flux through the top and bottom boundaries fluctuates in time and space (e.g., Tobie et al., 2003). For our analysis, we calculated the local heat flux from the thermal gradient at the boundary and spatially and temporally averaged this heat flux to obtain the mean heat flux. The heat-flux calculations were performed after the flows reached a statistical equilibrium (after the initial start-up transient was finished). The temporal averaging was performed using output frames over , , and yearperiods for η 0 = 10 13,10 14, and Pa s, respectively. These averaging times are all longer than the convective overturn times of years for η 0 = Pa s. The ConMan code assumes that the ice-layer thickness δ is constant throughout each simulation, so there is no direct way to account for ice-shell thickness fluctuations in response to thermal perturbations. Therefore, we cannot explicitly simulate a convective-conductive switch with the associated ice-shell thickness changes within any given ConMan simulation. However, we can perform constantthickness simulations to determine the equilibrium heat flux for a given shell thickness and tidal-flexing amplitude; we then use these results to infer how the thickness will change if the basal heat flux or tidal-flexing amplitude change. This approach is valid as long as the perturbations in heat flux or tidal-flexing amplitude occur on timescales long compared to the convective timescale τ conv, which is years for viscosities of Pa s. Even near the critical Rayleigh number, where the equilibration times are longer because of the smaller role of advection relative to conduction, the heat fluxes in our simulations equilibrate within years for melting-point viscosities of Pa s. In contrast, the expected changes in shell thickness associated with coupled orbital geophysical feedbacks are years (e.g., Ojakangas and Stevenson, 1986; Showman et al., 1997; Hussmann and Spohn, 2004). Therefore, the constant-thickness simulations provide an approximately correct description of the convection, at least for η Pa s. 3. Results and discussion 3.1. Basic morphology of the convection In agreement with earlier investigations (McKinnon, 1999; Wang and Stevenson, 2000; Sotin et al., 2002; Barr and Pappalardo, 2003; Showman and Han, 2004; Tobie et al., 2003), our results demonstrate that convection can occur under conditions relevant to Europa (Figs. 2 and 3). For an ice shell without internal heating, convection can occur in ice shells thicker than 10 and 21 km for meltingtemperature viscosities of and Pa s, respectively. In a tidally heated ice shell with tidal-flexing amplitude ε , convection can occur in layers thicker than 9 and 19 km for melting-temperature viscosities of and Pa s, respectively. The critical Rayleigh number Ra cr depends on tidal-heating rate, and is smaller when Fig. 2. Temperature field of a numerical simulation without internal heating in the ice shell. The domain is 25 km thick and 100 km wide. The melting point viscosity is Pa s and the basal Rayleigh number is The time of this frame is yr.

6 S (05) /FLA AID:7639 Vol. ( ) [DTD5] P.6 (1-14) YICAR:m5 v 1.39 Prn:25/05/2005; 7:41 yicar7639 by:diana p. 6 6 G. Mitri, A.P. Showman / Icarus ( ) Fig. 3. Temperature field of a numerical simulation with internal heating in the ice shell. The domain is 45 km thick and 180 km wide. The melting point viscosity is Pa s and the basal Rayleigh number is The tidal flexing amplitude of the ice shell is ε 0 = The time of this frame is yr. the tidal heating is larger. As expected, the temperaturedependence of the viscosity leads to formation of a 4 8-km thick stagnant lid at the surface; in contrast, the boundary layer at the bottom of the system is only 1 2-km thick and actively participates in the convection (Solomatov, 1995; Moresi and Solomatov, 1995; McKinnon, 1999; Deschamps and Sotin, 2001; Solomatov and Moresi, 2000; Showman and Han, 2004; Tobie et al., 2003). ForRa < , depending on the tidal-heating rate, the convection is steady-state, with periodic, regularly spaced upwellings and downwellings (Fig. 2). For Rayleigh numbers greater than this value, the convection is time dependent and lacks the regular symmetry of the steady-state regime (Fig. 3). The bottom panel of Fig. 3 illustrates that the tidal-heating rate depends on temperature. For η 0 = Pa s, the magnitude of tidal heating is greatest along the base of the stagnant lid and within cold descending plumes (Fig. 3b); in contrast, for η 0 = Pa s the greatest heating occurs at the bottom of the ice shell and in hot ascending plumes (e.g., Wang and Stevenson, 2000; Sotin et al., 2002) Nusselt number Rayleigh number relationship The Nusselt number, defined as Nu = Fδ k T, (11) where F is the heat flux, provides a measure of how vigorously the convection transports heat. Analytical investigations of internally or basally heated fluids suggest a relationship between the Nusselt number Nu and the Rayleigh number Ra i, evaluated using a characteristic viscosity in the interior of the convective sublayer, given by (Morris and Canright, 1984; Fowler, 1985) Nu = αra i β θ γ, (12) where θ ln η/ T characterizes the temperature-dependence of the viscosity. The coefficients α, β, and γ are dimensionless constants that depend on the boundary conditions of the convective system (Turcotte and Oxburgh, 1967; Roberts, 1979; Howard, 1964; Somerscales and Gazda, 1968; Dropkin and Somerscales, 1965; Goldstein et al., 1990). For the steady-state convection regime at low Rayleigh number, Dumoulin et al. (1999) suggest parameter values of α = 1.99, β = 1/5, and γ = 1, and for the time-dependent regime at high Rayleigh number α = 0.52, β = 1/3, and γ = 4/3. Between the stationary and the time-dependent regimes, there is a time-dependent transitional regime that has parameters α, β, and γ equal to the values for the stationary regime. These values are generally consistent with values suggested by Moresi and Solomatov (1995), Davaille and Jaupart (1993), and others; see Solomatov and Moresi (2000) for a summary.

7 S (05) /FLA AID:7639 Vol. ( ) [DTD5] P.7 (1-14) YICAR:m5 v 1.39 Prn:25/05/2005; 7:41 yicar7639 by:diana p. 7 Europa convection 7 Fig. 4. Relation between the basal Rayleigh number Ra and the Nusselt number Nu for an ice shell without internal heating. The figure shows a finite amplitude jump of Nusselt number at the critical Rayleigh number ( ). The relationship between Ra and Nu has two different trends for low and high Ra. Fig. 4 shows the Nu Ra relationship obtained from a broad range of our simulations. The figure illustrates that at Ra 10 9, a break in the slope occurs; the power-law exponent β is about 1/3 forra > 10 9 and 1/5 for < Ra < These results broadly agree with the theories summarized above. The β 1/5 regime encompases both steady-state convection at low Ra (< )aswell as time-dependent convection at higher Ra (> ); in other words, the transition between steady-state and timedependent convection occurs at lower Rayleigh number than the transition between the β 1/5 and β 1/3 regimes. Interestingly, the transition between the β 1/5 and 1/3 regimes occurs at Nu 5, which agrees with the results of Dumoulin et al. (1999). However, we expect that the value of Ra at this transition depends on the viscosity contrast; greater viscosity contrasts imply a lower value of Ra at the β 1/5 1/3 transition (Dumoulin et al., 1999). At Rayleigh numbers less than the critical Rayleigh number of , the shell loses heat purely by conduction, corresponding to Nu = 1. In the β 1/3 regime, the heat flux is essentially independent of ice-shell thickness, which implies that the iceshell thickness is extremely sensitive to perturbations in the basal-heat flux. In this regime, infinitesimal decreases in the heat flux supplied to the base of the shell cause complete refreezing of the shell, whereas infinitesimal increases in the heat flux supplied to the base of the shell cause thinning sufficient to force the shell into the β 1/5 regime (whereupon the ice-shell thickness is no longer so sensitive to basalheat flux). Given the observational evidence for an ocean (Kivelson et al., 2000), we therefore infer that the ice shell lies within either the β 1/5 convective regime or the conductive regime. The extreme thickness variations in the coupled orbital thermal parameterized-convection simulations of Hussmann and Spohn (2004) probably result from their choice of β = 0.3. In future studies, it may be preferable to use β = 1/5, which would probably lead in the convective regime to more modest ice-shell thickness variations than found by Hussmann and Spohn (2004). Nevertheless, we will show that substantial thickness variations can still accompany switches between convective and conductive states near the critical Rayleigh number. Interestingly, we find that immediately above the critical Rayleigh number, Nu increases discontinuously to 1.7, which implies that the convection jumps directly into a finite-amplitude regime (i.e., the convected flux is 70% of the conducted flux, rather than an infinitesimal fraction as would be predicted by linear theory). This result agrees with laboratory experiments using glycerol conducted by Stengel et al. (1982), who found that the Nusselt number at the critical Rayleigh number ranged from 1.1 to 1.2 for viscosity ratios of (and an extrapolation of his results to viscosity contrasts of 10 6 suggests Nu = 1.7 ± 0.5 at the critical Rayleigh number, consistent with our results). We propose the following explanation. At constant viscosity, linear theory predicts broad, domain-filling convection cells, and at weakly supercritical Rayleigh numbers, these cells produce only a minor perturbation to the conductive temperature profile. This behavior implies that a single convection cell, in the linear regime, spans temperatures ranging from that at the bottom to that at the top of the domain (i.e., small-amplitude convection cells are far from isothermal). When the viscosity depends strongly on temperature, however, convection cells cannot a span broad range of temperature, because only the viscosity at the lower boundary is low enough for fluid deformation to occur. This phenomenon initially inhibits convection (leading to higher critical Rayleigh numbers for temperature-dependent-viscosity than for constant-viscosity fluids). For convection to finally begin, the convection cells must immediately jump into an approximately isothermal state with a temperature close to that at the lower boundary. This is a finite-amplitude state, because finite temperature advection must counteract conduction to maintain the isothermal nature of the convection cells. The resulting Nusselt number depends on the thickness of the cells and in turn the viscosity contrast; if the convection cells fill one-third of the domain, the stagnant lid would be two-thirds of the domain thickness and the resulting Nusselt number would be 1.5. These arguments are broadly consistent with convective scaling theories, which predict finite velocities at the critical Rayleigh number when the viscosity contrast is large (e.g., Solomatov and Moresi, 2000) Conductive convective transitions and sensitivity of ice shell to thermal perturbations We now describe the dependence of heat flux on shell thickness and discuss the implications for Europa s thermal evolution. Fig. 5 illustrates how the heat flux depends on iceshell thickness for a sequence of runs with no tidal heating and the parameter values listed in Table 1. Each point rep-

8 S (05) /FLA AID:7639 Vol. ( ) [DTD5] P.8 (1-14) YICAR:m5 v 1.39 Prn:25/05/2005; 7:41 yicar7639 by:diana p. 8 8 G. Mitri, A.P. Showman / Icarus ( ) Fig. 5. Relation between the heat flux generated in the silicate mantle and the thickness of the ice shell (in a range 0 60 km) for ice melting-temperature viscosities of Pa s (diamonds), Pa s (triangles), and Pa s (squares). These numerical simulations are without internal heating in the ice shell. The radiogenic heat flux ranges from W m 2 (Melosh et al., 2004; Tobie et al., 2003). The maximum radiogenic heat flux is plotted in dashed line. resents the results of a simulation; diamonds, triangles, and squares correspond to simulations with η 0 = 10 13,10 14, and Pa s, respectively. Because tidal heating was excluded here, the time-averaged fluxes through the top and bottom boundaries are equal. In each case, the plotted flux corresponds to that needed to reach an approximate equilibrium (i.e., statistical steady state) under the constraints of constant ice-shell thickness and bottom-to-top temperature contrast. The solid curve shows the conductive heat flux F = k T/δ and the dotted line provides a schematic upper limit to the expected radiogenic heat flux from Europa s interior. (We adopt constant k here for consistency with ConMan, which assumes constant conductivity.) The figure illustrates several important points. First, the thinnest-shell simulations for each value of η 0 had subcritical Rayleigh numbers and produced conductive solutions; these simulations fall along the solid curve. Simulations that produced convection had total fluxes exceeding the conductive flux; these simulations fall above the solid curve. Second, the heat-flux discontinuity at the critical Rayleigh number is clearly evident; the total flux at Rayleigh numbers just slightly greater than Ra cr is about 1.7 times the corresponding conductive flux. Third, the heat fluxes for the convective cases decline with increasing shell thickness (despite the fact that Nu increases with Ra; see Fig. 4). Together, these traits imply that for a range of heat fluxes ( W m 2 for a melting-temperature viscosity of Pa s), there are two solutions for a given heat flux: one corresponding to a thin, conductive shell and the other to a thick, convective shell (see Fig. 1). This figure has important implications for the thermal evolution of Europa. A given point in Fig. 5 corresponds to a true equilibrium on Europa only if the heat flux transported through the shell equals that supplied from Europa s oceanic/silicate interior. If the heat flux from Europa s oceanic/silicate interior differs from the flux that can be transported at a given ice-shell thickness, then the iceshell thickness will change until the flux transported through the ice layer equals that supplied from below. For example, suppose the ice shell is 22-km thick, implying a heat flux through Europa s surface of 0.05 W m 2 for η 0 = Pa s (Fig. 5), and that tidal and radiogenic heating in the silicate interior supplies 0.03 W m 2 to the ocean. The mismatch of heat fluxes will cause freezing at the base of the shell, which will continue until the shell becomes 55-km thick. At this point the heat flux through Europa s surface reaches that supplied from below (0.03 W m 2 ;seefig. 5) and freezing ceases equilibrium is attained. These arguments based on Fig. 5 suggest that, in either branch of the curves where heat flux decreases with increasing shell thickness, the ice shell is stable: for a given heat flux supplied by the interior, decreases in ice-shell thickness away from the equilibrium cause increases in the heat flux transported by the shell, cooling, and hence thickening, which returns the shell to its equilibrium thickness. Increases in shell thickness away from the equilibrium cause decreases in heat flux transported by the shell, warming, and hence thinning, which again returns the shell toward its equilibrium thickness. Fig. 5 also implies that variations in heat flux from the silicate interior can cause large changes in ice-shell thickness. If the heat flux from the silicate interior changes by W m 2, then the equilibrium thickness of a convective shell changes by 7 km and 20 km for η 0 = and Pa s, respectively. In contrast, for a conductive ice shell, such a change in heat flux would cause only 1 2 km changes in the shell thickness. Near the critical Rayleigh number, modest variations in heat flux can force the ice shell to switch between conductive and convective regimes, with consequent rapid changes in the ice-shell thickness. Imagine a conductive ice shell in equilibrium with a large, but slowly declining, heat flux supplied from the silicate layer. The system would occupy a point along the solid curve at the upper left of Fig. 5 and would slowly slide down the curve as the flux decreased. The shell thickens accordingly, remaining conductive as long as the thickness is less than 10, 20, or 40 km for meltingtemperature viscosities of 10 13,10 14,or10 15 Pa s, respectively. Once the shell reaches this critical thickness, however, a dilemma arises: the continually declining flux implies that the shell must continue to grow, but any further increase in the shell thickness causes an increase, not a decrease, in the flux transported through the ice layer. The system can no longer remain in equilibrium: it becomes convective, but because the convected flux greatly exceeds that supplied from below, the shell thickness increases rapidly until the system reaches a new equilibrium. For η 0 = Pa s, the shell would jump from 10- to 35-km thickness, and for η 0 = Pa s, the shell would jump from 19- to 65-km thickness. This constitutes a switch from the conductive to the convective solution at a given heat flux. Suppose now that the heat flux supplied to the base began to increase. The shell would

9 S (05) /FLA AID:7639 Vol. ( ) [DTD5] P.9 (1-14) YICAR:m5 v 1.39 Prn:25/05/2005; 7:41 yicar7639 by:diana p. 9 Europa convection 9 remain on the convective branch, thinning in response to the increased flux. Eventually, the shell would reach the critical Rayleigh number (at a thickness of 12, 22, or 45 km for melting-temperature viscosities of 10 13,10 14,or10 15 Pa s, respectively); further thinning would force the shell into the conductive regime. However, this involves another crisis, because further thinning causes a decrease, not an increase, in the transported flux, forcing the system away from equilibrium (it cannot transport all of the flux supplied from below). The shell would then thin rapidly until a new equilibrium is attained. For η 0 = Pa s, the shell would jump from 10- to 6-km thickness, and for η 0 = Pa s, the shell would jump from 19- to 14-km thickness. Interestingly, the switches are asymmetric: conductive-to-convective switches involve much larger changes in the ice-shell thickness than convective-to-conductive switches. We emphasize that these rapid conductive convective switches result directly from the discontinuity in heat flux at Ra cr and, to our knowledge, have never been discussed to date for icy satellites. Many icy-satellite evolution models have been published, but these generally adopt Nu Ra relationships (or equivalently, F δ relationships) that are continuous at the critical Rayleigh number (e.g., Spohn and Schubert, 2003; Deschamps and Sotin, 2001; Hussmann et al., 2002; Hussmann and Spohn, 2004; Showman et al., 1997; Kirk and Stevenson, 1987; and others). Therefore, these studies explicitly preclude the conductive convective switches discussed here. Fig. 6 shows the heat flux through the bottom and top boundaries vs shell thickness for models that include tidal heating. For an internally heated ice shell, the basal flux F hbl depends more strongly on δ than for a shell without internal heating, which implies that variations in the flux supplied from the silicate interior cause smaller changes in shell thickness for a tidally heated shell (at constant tidalflexing amplitude) than for a shell with zero tidal heating. For a melting-temperature viscosity of Pa s and tidalflexing amplitude of , for example, a basal heat-flux variation of W m 2 would cause a change in shell thickness of 1 4 km for a convecting shell, depending on the thickness (Fig. 6a). For a melting-temperature viscosity of Pa s and tidal-flexing amplitude of , such a heat-flux perturbation would alter the shell thickness by 6 km(fig. 6b). These thickness changes are a factor of 3 4 smaller than for a shell with no heating (Fig. 5). The heat-flux jump at the critical Rayleigh number remains strong when tidal heating is included in the simulations (Fig. 6), which implies that conductive convective transitions can occur in tidally heated shells just as they can in shells without tidal heating. If heat-flux variations from the silicate interior occur at constant ice-shell tidal-flexing amplitude, then Fig. 6 can describe evolutionary paths followed by Europa. For example, suppose Europa starts with a thin, conductive shell in equilibrium with a large basal-heat flux that gradually declines in time. The shell will gradually thicken in the conductive regime and then will switch Fig. 6. Relation between the heat flux calculated in the hot boundary layer of the convective ice shell (triangles) and in the stagnant lid (diamonds) vs the thickness of the ice shell δ for simulations that include tidal heating in the ice shell. Solid curve shows the relationship between heat flux F and δ for a conductive ice shell without internal heating. The radiogenic heat flux ranges from to W m 2 (Melosh et al., 2004; Tobie et al., 2003). The maximum radiogenic heat flux is plotted in dashed line. The critical thickness of the ice shell, corresponding to the critical Rayleigh number, is 9 and 18 km in the top and bottom panels, respectively. (a) Melting-temperature viscosity is Pa s and the tidal-flexing amplitude is ε 0 = (b) Melting-temperature viscosity is Pa s and the tidal flexing amplitude is ε 0 = rapidly to a much thicker convective regime (from 8- to 15- km thickness in Fig. 6a and from 18- to 31-km thickness in Fig. 6b). Similarly, a gradually increasing basal flux in the convective regime would lead to gradually thinner convective layers until a rapid switch to a much thinner conductive state occurred (from 9- to 5-km thickness in Fig. 6a and 19- to 11-km thickness in Fig. 6b). Fig. 7 shows the dependence of the bottom and top fluxes transported by the ice shell on the tidal-flexing amplitude ε 0. Stronger tidal flexing leads to increased flux through Europa s surface but decreased flux into the ice layer from below. This behavior occurs because the ice shell is warmer when the tidal flexing is greater, which enhances loss of heat through the top surface and inhibits gain of heat through the bottom surface. At high Rayleigh numbers, the heat fluxes

10 S (05) /FLA AID:7639 Vol. ( ) [DTD5] P.10 (1-14) YICAR:m5 v 1.39 Prn:25/05/2005; 7:41 yicar7639 by:diana p G. Mitri, A.P. Showman / Icarus ( ) Fig. 8. Ice-shell thickness as a function of the heat flux from the interior of Europa and the tidal flexing amplitude of the ice shell for η 0 = Pa s. Dashed and dotted lines correspond to convective and conductive solutions, respectively. Shown are 10-km convective (filled circles), 15-km convective (diamonds), 30-km convective (triangles), 45-km convective (squares), 8-km conductive (crosses), 7-km conductive (asterisks), and 5-km conductive (pluses) solutions. The dotted line at the top shows the conductive solution for a 3-km shell. The gray area denotes the region where both conductive and convective solutions exist for a given tidal-flexing amplitude and basal heat flux. Variations in these two parameters can cause rapid switches between conductive and convective states; see text. Fig. 7. Heat flux through the hot boundary layer of the convective ice shell (triangles) and stagnant lid (diamonds) versus the tidal flexing amplitude ε 0 for tidally heated ice shells. Increasing the orbital eccentricity and, consequently, the tidal-flexing amplitude decreases the ability of the ice shell to receive conducted heat from the interior of the satellite but increases the flux lost to space. Melting-temperature viscosity is Pa s. (a) Ice-shell thickness is 15 km; (b) ice-shell thickness is 30 km; (c) ice-shell thickness is 45 km. into the base are time dependent, and the vertical bars illustrate the range of variation; the central point gives the timeaveraged flux. The fluctuations generally occur on a convective overturn timescale, years, which is much smaller than typical orbital and thermal evolution timescales of years. Fig. 8 summarizes the dependence of ice-shell thickness on the basal heat flux and ice-shell tidal-flexing amplitude for a range of conductive and convective simulations. The figure can be viewed as a contour plot of layer thickness; the dashed curves connect convective solutions for a given layer thickness, while dotted curves connect conductive solutions for a given layer thickness. The gray region denotes the parameter ranges where both conductive and convective solutions (with distinct ice-layer thicknesses) exist for a given tidal-flexing amplitude and basal heat flux. Above the gray region, only conductive solutions exist, and below the gray region, only convective solutions exist. The basal heat-flux (vertical axis) and ice-shell tidalflexing amplitude (horizontal axis) can be viewed as parameters whose values are approximately independent of the ice-shell state; the magnitude of radiogenic/tidal heating and secular cooling of Europa s silicate interior determine the former, while Europa s eccentricity and tidal Love number determine the latter, e.g. (Ojakangas and Stevenson, 1989; Moore and Schubert, 2000). As these parameters vary, the location of the system on Fig. 8 changes, and Fig. 8 then shows how the ice-shell thickness varies in time (as long as the change in basal heat-flux and tidal-flexing amplitude occur with time constants of > years; see Section 2). Within the gray region, which of the solutions Europa occupies depends on prior history. If the system enters the gray region from above, the shell will remain conductive, but if the system enters the gray region from below, the system will remain convective. Large-amplitude switches between con-

11 S (05) /FLA AID:7639 Vol. ( ) [DTD5] P.11 (1-14) YICAR:m5 v 1.39 Prn:25/05/2005; 7:41 yicar7639 by:diana p. 11 Europa convection 11 ductive and convective states occur when the lower boundary of the gray region is crossed from above or the upper boundary of the gray region is crossed from below; these correspond to conductive-to-convective and convective-toconductive switches, respectively. Conductive-to-convective switches cause increases in thickness ranging from 30 km at zero tidal heating to 8 km at a tidal-flexing amplitude of Interestingly, Fig. 8 shows that fluctuations of the basalheat flux at constant tidal-flexing amplitude are more likely than fluctuations of tidal-flexing amplitude at constant basalheat flux to produce convective conductive switches. In reality, fluctuations in basal-heat flux and tidal-flexing amplitude are probably correlated. Changes in basal-heat flux most plausibly result from changes in the tidal-heating rate in the silicate interior, and because the eccentricity is the primary factor controlling the tidal-heating rate, we expect that increases in basal-heat flux are probably accompanied by increases in ice-shell tidal-flexing amplitude (and vice versa). The system would then follow a path extending between lower-left and upper-right in Fig. 8. If Europa s interior composition matches carbonaceous chondrites, then the current radiogenic heat flux is about 0.01 W m 2, which is substantially below the parameter ranges that allow convective-conductive switches for η 0 = Pa s; however, if the total silicate-layer heat flux (tidal + radiogenic) is several times the radiogenic value, then factor-of-two changes in tidal-flexing amplitude and basal-heat flux may result in large-amplitude switches between the conductive and convective states. Tidal-flexing amplitude is proportional to eccentricity, so such switches may occur if Europa undergoes factor-of-two changes in eccentricity. Models for the coupled orbital-geophysical evolution of the Galilean satellites show that two-fold changes in eccentricity are quite plausible (e.g., Ojakangas and Stevenson, 1986; Showman and Malhotra, 1997; Showman et al., 1997; Hussmann and Spohn, 2004), which suggests the plausibility of large-amplitude conductive convective switches. Even in the absence of convective conductive switches, Fig. 8 shows the sensitivity of ice-shell thickness to fluctuations in heating. A variation in ε 0 from 0 to changes the ice-shell thickness by 30 km for a fixed basalheat flux of W m 2. A variation of the tidal flexing amplitude ε 0 from 1.0 to , and a simultaneous increase of the heat flux from the rocky interior from to W m 2, produce a variation in thickness of 30 km. The time scale of the conductive convective switches can be estimated as follows. When a switch occurs, the mismatch between the flux that can be transported through the ice shell and the flux supplied from below is F 0.02 W m 2 (Figs. 5, 6, and 8). This mismatch causes melting or freezing. The resulting timescale for the ice shell to achieve a new equilibrium is τ Lρ δ/ F, where δ is the variation in thickness and L = Jkg 1 is the latent heat. For a typical value of δ 10 km, the equilibration timescale is τ years. We find that the maximum heat flux that can be transported by convection is F max 0.06 W m 2 for η 0 = Pa s, F max W m 2 for η 0 = Pa s, and F max W m 2 for η 0 = Pa s (Figs. 5 and 6). These maximum fluxes are largely independent of tidalheating rate. For higher values of the heat flux, convection cannot occur in the ice shell and the heat is transported by thermal conduction. This conclusion agrees with the results of Showman and Han (2004). The minimum heat flux from the interior of Europa is the heat of radiogenic decay, which for carbonaceous chondritic abundances of Europa s rocky portion is just under 0.01 W m 2 (Spohn and Schubert, 2003). Ourresults show that, if no tidal heating occurs, the ice shell would eventually freeze if the melting-temperature viscosity is or Pa s. For a melting-temperature viscosity of Pa s, however, the shell can never freeze even if tidal-heating is zero in both the ice shell and silicate interior. As discussed previously, however, plausible tidalheating rates imply that the shell does not freeze completely even at melting-temperature viscosities of or Pa s; this result is consistent with the results of many previous studies (e.g., Spohn and Schubert, 2003; Tobie et al., 2003; Hussmann et al., 2002; Deschamps and Sotin, 2001) Satellite expansion/contraction caused by conductive convective switch The large variations in ice-shell thickness during a switch between conductive and convective configurations can cause rapid radial expansion of Europa. A refreezing of 10 km of the ice shell causes a radial expansion of 1 km, corresponding to a fractional satellite volume change of and a global, linear extensional strain of at the surface. Such an expansion would fracture the surface and generate km 2 of new surface area, which could potentially result in the formation of bands or other extensional tectonic structures. Here we estimate the stress in the ice shell during the radial expansion. The stress as a function of the ice-shell temperature is calculated using the flow law of Eq. (7) in the generalized Maxwellian viscoelastic equation ε = σ µ + σ η + Aσn d p exp ( Q RT ), (13) where µ = Pa is the rigidity, η is the Newtonian viscosity, and the third term represents non-newtonian deformation. The composite constitutive steady-state creep law proposed by Goldsby and Kohlstedt (2001) show that, at Europa s surface, dislocation and diffusion creep are predominant, so the final term here represents dislocation creep with n = 4 and Q = 60 kj mol 1 (see Table 2). We expect that satellite expansion does not depend on ice-shell stress, so we independently specify ε as a function of time (to represent satellite expansion) and solve the differential equation to find the maximum value of stress at a given temperature